arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2206.03347math.OCmath.APmath.FA
收藏

一般熵最优输运费用的收敛速度

Convergence rate of general entropic optimal transport costs

Guillaume Carlier, Paul Pegon, Luca Tamanini

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们研究的收敛速度的最佳熵成本$v_\varepsilon$的最优输运成本的噪声参数$\varepsilon \downarrow 0$。我们表明，对于一类大的成本函数$c$ $\mathbb{R}^d\times \mathbb{R}^d$（最佳计划不一定是唯一的或诱导的运输地图）和companimously支持和$L^{\infty}$边际，一个有$v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$。上界是通过块逼近策略和亚历山德罗夫定理的积分变体得到的。在$c$上的无穷小扭曲条件下，即$\nabla_{xy}^2 c(x,y)$的可逆性条件下，利用Minty的技巧，通过在$d$维中建立对偶间隙的二次分离，得到了该下界.

英文摘要：

We investigate the convergence rate of the optimal entropic cost $v_\varepsilon$ to the optimal transport cost as the noise parameter $\varepsilon \downarrow 0$. We show that for a large class of cost functions $c$ on $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on $c$, i.e. invertibility of $\nabla_{xy}^2 c(x,y)$, we get the lower bound by establishing a quadratic detachment of the duality gap in $d$ dimensions thanks to Minty's trick.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑